Rigidity and flexibility of group actions on the circle

Transcription

1 Rigidity and flexibility of group actions on the circle Kathryn Mann Abstract We survey rigidity results for groups acting on the circle in various settings, from local to global and C 0 to smooth. Our primary focus is on actions of surface groups, with the aim of introducing the reader to recent developments and new tools to study groups acting by homeomorphisms. Contents 1 Introduction 2 2 The ubiquity of surface groups Flat bundles and foliations Geometric actions on manifolds Representation spaces and character varieties An introduction to rigidity Local rigidity Full rigidity Strong (Mostow) Rigidity The Zimmer program Two perspectives on rigidity of group actions Rigidity from hyperbolicity Maximality and geometricity Not just surface groups Rotation number theory Poincaré s rotation number Semi-conjugacy The Euler number as a rotation number Rotation numbers as coordinates Rotation number as a quasimorphism Quasimorphisms and H 2 b (G; R) Rotation numbers of products The Calegari Walker algorithm Applications Rigidity of geometric representations Matsumoto s rigidity theorem Modifications for the general case Consequences of full rigidity Flexibility of group actions 34

2 1 Introduction Given a group and a manifold M, can one describe all actions of on M? Put in such broad terms this question is too ambitious, but certain special cases are quite tractable. In this paper we focus on the special case M = S 1, and eventually further specialize to the case where is the fundamental group of a closed surface. As should become clear, even this very special case leads to a remarkably rich theory, with deep connections to problems in topology, geometry and dynamics. Perhaps the reader is already familiar with the wonderful survey paper of Ghys, titled Groups acting on the circle [26]. If not, we recommend it highly both on its own and as a companion to this work. Although this survey also treats groups acting on S 1, we have chosen to take a rather di erent approach while Ghys starts by describing the dynamics of a single homeomorphism, and ultimately aims to prove that higher rank lattices do not act on on the circle, here we focus from the beginning on group actions, on groups that do act on S 1, and on groups that act in many di erent ways. Our aim is to explore and understand the rigidity and flexibility of these actions. Outline and scope. We begin by motivating the study of spaces of group actions, as well as our focus on surface groups. Section 3 introduces the concept of rigidity in various forms, and Section 4 presents some well-known examples. The remainder of the paper is devoted to advertising rotation number coordinates as a means of studying spaces of group actions on S 1, culminating in a description of recent results and new approaches to old problems, from [15], [42], and others. It is our hope that this paper will be both valuable and accessible to a wide audience. Whether you are a graduate student looking for an introduction to group actions, an expert with an interest in character varieties and geometric structures, or instead have a background in hyperbolic dynamics, there should be something here for you. Regretfully, there are many topics we have been forced to omit or gloss over in the interest of brevity and accessibility. These include regularity of actions, circular orders on groups, and bounded cohomology in fact, we have made the perhaps unconventional choice to keep discussions of group cohomology separate and in the background. Of course, in all cases we have given references wherever possible, and hope this paper serves as a welcoming entry point for the interested reader. Acknowledgements. Thanks to all who have shared their perspectives on (surface) groups acting on the circle, especially Christian Bonatti, Danny Calegari, Benson Farb, Étienne Ghys, Shigenori Matsumoto, Andrés Navas, Maxime Wol, and the members of the Spring 2015 MSRI program Dynamics on Moduli Spaces of Geometric Structures. Thanks additionally to Subhadip Chowdhury, Hélène Eynard-Bontemps, Bena Tshishiku, and Alden Walker for comments on the manuscript, and to the organizers and participants of Beyond Uniform Hyperbolicity 2015 where a portion of this work was presented as a mini-course. 2 The ubiquity of surface groups We begin by introducing a few of the many ways that actions of finitely generated groups come up in areas of topology, geometry and dynamics; with an emphasis on surface groups acting on S 1. We ll see actions of surface groups on S 1 arise as the most basic case of flat or foliated bundles, as the essential examples of geometric actions on the circle, and through analogy with the study of character varieties. In addition to illustrating the diversity of questions related to surface groups acting on the circle, this section also covers basic background material used later in the text. Convention 2.1. Surface group means the fundamental group of a closed, orientable surface of genus g 2. 2

3 Convention 2.2. For convenience, we assume throughout this work that everything is oriented: all manifolds are orientable, bundles are oriented, foliations are co-oriented, and homeomorphisms preserve orientation. We use Di r (M) to denote the group of C r orientation-preserving di eomorphisms of a manifold M, and Homeo(M) =Di 0 (M) the group of orientation-preserving homeomorphisms. 2.1 Flat bundles and foliations Let M and F be manifolds of dimension m and n respectively, and E a smooth F bundle over M. ThebundleE is called flat if any of the following equivalent conditions hold: i) E admits a connection form with vanishing curvature. ii) E admits a smooth foliation of codimension n transverse to the fibers. iii) E admits a trivialization with locally constant transition maps (i.e. totally disconnected structure group) The foliation of condition ii) is given by the integral submanifolds of the (completely integrable) horizontal distribution given by the connection form, and is defined locally by the leaves U {p} of the trivialization U F! U M from condition iii). Viewing the foliation or trivialization as the central object, rather than the connection form, allows one to extend the notion of flat to bundles of lower regularity. Definition 2.3. A (topological) flat F bundle over M is a topological F bundle with a foliation of codimension n transverse to the fibers. Note that this definition requires only C 0 regularity to be transverse to the fibers is a statement about the existence of local charts to R m R n, with fibers mapped into to sets of the form R n {y} and leaves of the foliation to {x} R m. In this case, we define the regularity of the flat bundle to be the regularity of the foliation. Holonomy and group actions. The relationship between flat bundles and groups acting on manifolds is given by the holonomy representation, which we describe now. Fix a basepoint x 2 M, and an identification of F with the fiber over x. Theholonomy of a flat bundle of class C r is a homomorphism : 1 (M,x)! Di r (F ) defined as follows: Given p 2 F, and a loop :[0, 1]! M based at x and representing an element of 1 (M,x), there is a unique lift of to a horizontal curve :[0, 1]! E with (0) = p and (1) 2 F. Define ( )(p) := (1). One checks easily that ( )(p) depends only on the based homotopy class of, that ( ) is a homeomorphism of F, and is a homomorphism 1 (M,x)! Di r (F ). Conversely, given a homomorphism : 1 (M,x)! Di r (F ), one can build a flat bundle with holonomy as a quotient of the trivial F bundle over the universal cover M. The quotient is ( M F )/ 1 (M) where 1 (M) acts diagonally on M F by deck transformations on M and by on F ;the foliation descends from the natural foliation of M F by leaves of the form M {p}. These constructions produce a one-to-one correspondence {Flat F bundles of class C r } holonomy! {representations 1 (M,x)! Di r (F )}. This is more often stated as a correspondence between flat bundles up to equivalence and representations up to conjugacy in Di r (F ). Later, in the special case where F = S 1, we will also consider the equivalence relation on flat bundles that corresponds to semi-conjugate representations. Very roughly speaking, 3

4 semi-conjugate flat bundles are obtained by replacing a leaf L with a foliated copy of L I (or replacing many leaves by foliated product regions), and/or by the inverse operation of collapsing foliated regions down to single leaves. From this perspective, questions about rigidity and flexibility of representations are questions about the (local) structure of the space of horizontal foliations on a given fiber bundle. We ll begin to discuss these questions in Section 3. For now, we focus on an even more fundamental problem: Basic Problem 2.4. Which bundles admit a foliation transverse to the fibers? Equivalently, which bundles can be obtained from a homomorphism : 1 (M)! Di r (F ) as described above? Despite its seemingly simple statement, the only nontrivial case in which Problem 2.4 has a complete answer is when the fiber F is S 1 and base M is a surface. This answer is given by the Milnor Wood inequality. Characterizing flat bundles: The Milnor Wood inequality. Convention 2.5. For the rest of the text, g denotes the closed surface of genus g, assuming always that g 2. The Milnor Wood inequality characterizes which circle bundles over g admit a foliation transverse to the fibers in terms of a classical invariant called the Euler number. Theorem 2.6 (Milnor Wood inequality, [47] [60]). A topological S 1 bundle over g admits a foliation transverse to the fibers if and only if the Euler number e of the bundle is less than or equal to ( g ) in absolute value. We will give an elementary definition of the Euler number in Section 5.3 and eventually give a proof of the Milnor Wood inequality as well 1. As for bundles over surfaces with fibers other than S 1, the question of which bundles admit horizontal foliations is quite di cult, even when the fiber is a surface. As an example, Kotschick and Morita [39] recently gave examples of flat surface bundles over surfaces with nonzero signature, but whether various other characteristic classes can be nonzero on flat bundles remains open. 2.2 Geometric actions on manifolds Our second motivation for studying surface group actions on the circle comes from the notion of a geometric action on a manifold. Recall that a cocompact lattice in a Lie group G is a discrete subgroup such that G/ is compact. Definition 2.8. Let be a finitely generated group, and M a manifold. A representation :! Homeo(M) is called geometric if it is faithful and with image a cocompact lattice in a transitive, connected Lie group G Homeo(M). This definition is motivated by Klein s notion of a geometry as a pair (X, G), where X is a manifold and G a Lie group acting transitively on X. In the case where M is a manifold of the same dimension as X, ageometric structure or (G, X) structure on M, in the sense of Klein, is specified by a representation 1 (M)! G Homeo(X). As a basic example, any compact surface 1 For impatient readers who happen to be familiar with classifying spaces, here is a quick definition of the Euler number to tide you over. Definition 2.7. Let E be a topological S 1 bundle over a surface. There is an isomorphism H 2 (B Homeo(S 1 ); Z) = H 2 (B SO(2); Z) = Z induced by the inclusion of SO(2) into Homeo(S 1 ). Thus, the Euler class for SO(2) bundles in H 2 (B SO(2); Z) (a generator) pulls back to a generator of H 2 (B Homeo(S 1 ); Z). The (integer) Euler class of E in H 2 ( ; Z) is the pullback of this element under the classifying map. The Euler number is the integer obtained by evaluating the Euler class on the fundamental class of. 4

5 of genus g 2 admits a hyperbolic structure in fact, many di erent hyperbolic structures and these correspond roughly 2 to the conjugacy classes of discrete, faithful representations : 1 ( g )! PSL(2, R) =Isom(H 2 ). The correspondence comes from considering g = H 2 / ( g ) with the induced hyperbolic metric. But this is a bit of a digression, at the moment we are more interested in the action of PSL(2, R) on the circle... Geometric actions on S 1. Specializing to the situation M = S 1, it is not hard to give a complete classification of the connected Lie groups acting transitively on M. These are SO(2), the group of rotations; PSL(2, R), the group of Möbius transformations; and the central extensions PSL (k) of PSL(2, R) of the form 0! Z/kZ! PSL (k)! PSL(2, R)! 1. The natural action of PSL (k) on S 1 comes from taking all lifts of Möbius transformations of S 1 to the k-fold cover of S 1 (which is, conveniently, also a circle). As a concrete example, PSL (2) = SL(2, R) with its standard action on S 1 considered as the space of rays from the origin in R 2. A nice proof of this classification of Lie subgroups can be found in [26, Sect. 4.1]; the essential observation originally due to Lie is that a Lie group acting faithfully on a 1-manifold can have at most a 3-dimensional Lie algebra. Given this classification, it is only a small step to describe all geometric actions. Theorem 2.9 (Geometric actions on S 1 ). Let :! Homeo(S 1 ) be a geometric action. Either i) is finite cyclic, and is conjugate to a representation into SO(2), or ii) is a finite extension of a surface group, and is conjugate to a representation with image in PSL (k). In other words, up to finite index, all geometric actions on S 1 come from surface groups. Proof. The discrete, cocompact subgroups of SO(2) are exactly the finite cyclic groups. Suppose now instead that H is a cocompact subgroup of PSL (k). The image of H under projection to PSL(2, R) is a discrete, cocompact subgroup of PSL(2, R), so contains a surface group as finite index subgroup. It follows that H is a subgroup of the central extension of this surface group by Z/kZ and hence itself a finite extension of a surface group. There are many examples of geometric actions of surface groups on the circle. The general construction is as follows. Example 2.10 (Surface groups in PSL (k) ). Let g = 1 ( g ) with standard presentation ha 1,b 1,...a g,b g Q i [a i,b i ]i. Let : g! PSL(2, R) be a discrete, faithful representation. For each generator a i and b i of g, pickalift (a i ) of (a i ) and (b i ) of (b i )topsl (k). One can check directly that the relation Q i [ (a i), (b i )] = id is satisfied precisely when k divides 2g 2; in this case, we have defined an action of g on the (k-fold covering) circle by homeomorphisms in PSL (k). Remark 2.11 (for the experts...). Example 2.10 can be nicely rephrased in more cohomological language: the obstruction to lifting an action of g on S 1 to the k-fold cover is exactly divisibility of the Euler class by k, as the lifted action gives a flat bundle which is the k-fold fiber-wise cover of the original. Faithful representations of g with image a lattice in PSL(2, R) correspond to bundles topologically equivalent to the unit tangent bundle of g, which has Euler number ± ( ), depending on orientation. We ll see that geometric actions are one source of rigidity, and in the case of actions of surface groups on S 1, conjecturally the only source of a strong form of C 0 rigidity (see Theorem 4.17 and Conjecture 4.18 below). 2 Technically, it is marked hyperbolic structures that correspond to conjugacy classes of discrete, faithful representations. 5

6 2.3 Representation spaces and character varieties Closely related to the theme of geometric actions is the classical study of representation spaces and character varieties. This subject also has particularly strong ties to surface groups. Definition Let be a discrete group, and G a topological group. The representation space Hom(,G) is the set of homomorphisms :! G, with the topology of pointwise convergence. In this topology, n converges to if n ( ) converges to ( )ingfor each 2. This is equivalent to requiring convergence of n (s) for each s in a generating set for. It is sometimes also called the algebraic topology on Hom(,G). The character variety X(,G) is, roughly speaking, the quotient of Hom(,G) bythe conjugation action of G. To explain this terminology (why variety? why only roughly?), we detour for a minute into the special case where G is an linear algebraic group. In this case, Hom(,G) has the structure of an algebraic variety whenever is finitely presented. Concretely, if S is a finite generating set for,thenhom(,g) can be realized as a subset of G S cut out by finitely many polynomial equations coming from the relators in. Unfortunately, the quotient Hom(,G)/G is not a variety in general the quotient is not even Hausdor. However, there is a better way to take a kind of algebraic-geometric quotient (the details of which we will not discuss here) and the resulting (Hausdor, nice) space is called the character variety. In many cases, it can genuinely be identified with the variety of characters of representations! G. See e.g. [54] for a thoughtful introduction to the special case G = PSL(2, C), and [40] for a more general treatment. Character varieties of surface groups. There are a number reasons why character varieties of surface groups are particularly interesting, many coming from the perspective of moduli spaces of geometric structures. For example, we mentioned above the relationship between discrete, faithful representations g! PSL(2, R) and hyperbolic structures on g. There are analogous relationships between convex real projective structures and certain representations g! PGL(3, R), between Higgs bundles on Riemann surfaces and representations to SL(n, C), etc. 3 Representation spaces of surface groups are also interesting in their own right for example, they admit a natural symplectic structure (discovered by Atiyah and Bott for spaces of representations into compact Lie groups, and Goldman in the noncompact case). We cannot possibly do justice to the theory here, and instead refer the reader to the recent book of Labourie [40]. However, it will be convenient for us to say a few words about the basic motivating case Hom( g, PSL(2, R)). For the sake of simplicity, we will discuss only representation spaces and not character varieties. The structure of Hom( g, PSL(2, R)). As PSL(2, R) is an algebraic group, and g finitely presented, Hom( g, PSL(2, R)) is a real algebraic variety. In particular, this implies that Hom( g, PSL(2, R)) has only finitely many components. How many? The Euler number of representations can be used to give a lower bound. By the Milnor Wood inequality (Theorem 2.6), there are 4g 3 possible values of the Euler number of a representation 2 Hom( g, PSL(2, R)), ranging from ( g )to ( g ). It s not hard to construct examples to show that each value in this range is attained: in brief, a hyperbolic structure on g defines a representation with Euler number 2g + 2, a singular hyperbolic structure with two order 2 cone points has Euler number 2g + 1, and every other negative value can then be constructed by a surjection g! h followed by a representation h! PSL(2, R) as above. Positive values can then be attained by conjugating by an orientation-reversing map of S 1. We ll also see later that the Euler number varies continuously, and hence is constant on connected components of Hom( g, PSL(2, R)). This means that Hom( g, PSL(2, R)) has at least 3 See [30] for a more detailed description of the relationships between moduli spaces and character varieties 6

7 4g 3 connected components. A remarkable theorem of Goldman states that this easy bound is sharp. 3 connected components, com- Theorem 2.13 (Goldman [29]). Hom( g, PSL(2, R)) has 4g pletely classified by the Euler number. The situation is quite di erent when G is not algebraic. For example, Question 2.14 (Open). Does Hom( g, Homeo(S 1 )) have finitely many connected components? What about Hom( g, Di r (S 1 )), for any r? Only very recently was it shown that the Euler number does not classify connected components of Hom( g, Homeo(S 1 )). A particular consequence of the work in [42] and [5] gives the following. Theorem 2.15 ([42]; [5] for the case G = Di 1 (S 1 )). Let G = Di r (S 1 ) for any 0 apple r apple1. There are more than 2 g connected components of Hom( g,g) consisting of representations with Euler number g 1. The essential ingredient in Theorem 2.15 is a rigidity result (Theorem 4.17 below), which is one of the primary motivations of this paper. Theorem 4.17 in fact says much more about the number of connected components of Hom( g,g) see Theorem 7.7 below. We ll explain the theorem and its proof in section 7. Unfortunately, the proof does not seem to give a hint at whether Question 2.14 has a positive or negative answer! Remark 2.16 (Regularity matters). Although the statement of Theorem 2.15 is uniform over all G = Di r (S 1 ), i.e. independent of r, one generally expects the topology of Hom( g, Di r (S 1 )) to depend on r. For instance, for the genus 1 surface group Z 2,wehave Easy theorem: Hom(Z 2, Homeo(S 1 )) is connected. Highly nontrivial theorem: Hom(Z 2, Di 1 (S 1 )) is path connected (Navas, [51]). Open question: is Hom(Z 2, Di 1 (S 1 )) locally (path) connected? Open question: is Hom(Z 2, Di r (S 1 )) connected, for any r 2? See [4] for discussion and progress on the Hom(Z 2, Di 1 (S 1 )) case, and [50] for many discussions of (often quite subtle) issues of regularity of group actions on the circle. Infinite dimensional Lie groups? Goldman s proof of Theorem 2.13 relies heavily on the Lie group (i.e. manifold) structure of PSL(2, R). It is conceivable, though perhaps optimistic, that some of his strategy might carry over to the study of representations to Di r (S 1 ). For any manifold M, and r 1, the group Di r (M) has a natural smooth structure (as a Banach manifold, or Fréchet manifold for the case of Di 1 (M)), and the tangent space at the identity can be identified with the Lie algebra of C r vector fields on M, making Di r (M) an infinite dimensional Lie group. In the case M = S 1, the structure of the Lie algebra Vect(S 1 )isquite well understood. Although this doesn t carry over for Homeo(M), there are good candidates for the Lie algebra of homeomorphisms of the circle, see e.g [41]. 3 An introduction to rigidity We turn now to the main theme of this work, rigidity of group actions on S 1. We begin by introducing several notions of rigidity, ranging from local to global. 3.1 Local rigidity Loosely speaking, an action of a group is rigid if deformations of this action are trivial in some sense, usually taken to mean that they are all conjugate. There are several ways to formalize this notion, for example: 7

8 Rigidity Definition 3.1 (Local rigidity). Let G be a topological group. A representation :! G is locally rigid if there is a neighborhood U of in Hom(,G) such that each 0 2 U is conjugate to in G. 4 In the special case G = Di 1 (M), there is a stronger version of local rigidity called di erentiable rigidity. Here, the conjugacy has higher regularity than expected. Rigidity Definition 3.2 (Local di erentiable rigidity, classical sense). A representation :! Di 1 (M) has local di erentiable rigidity if every representation in Hom(, Di 1 (M)) su ciently close to in the C 1 topology on Di 1 (M) is (smoothly) conjugate to. There are many possible variants of Definition 3.2 one can replace C 1 and smooth with C r and C s, one can ask only that representations in Hom(, Di 1 (M)) close to and C 0 conjugate to are smoothly conjugate to, etc. In the opposite direction, structural stability is a notion of rigidity where the conjugacy is less regular than the perturbation. Classically, a flow f t on manifold M is said to be structurally stable if every su ciently small C 1 perturbation g t of the flow is topologically equivalent to f t, meaning that there is a homeomorphism of M mapping flowlines of g t to flowlines of f t.a natural adaptation of this definition to group actions is the following. Rigidity Definition 3.3 (Structural stability). A representation :! Di 1 (M) isstructurally stable if there is a neighborhood U of in Hom(, Di 1 (M)) consisting of representations C 0 conjugate to. In the next section we ll give two proofs of structural stability for actions of surface groups on the circle. Deformations in Homeo(S 1 ). In the case where G = Homeo(S 1 ), there is another reasonable notion of trivial deformation of an action. This comes from a construction of Denjoy. Construction 3.4 ( Denjoy-ing an action). Let be a countable group, and 2 Hom(, Homeo(S 1 )). Enumerate the points of an orbit ( )x = {x 1,x 2,x 3,...}, and modify S 1 by replacing each point x i with an interval of length t2 i to get a new circle S t.thereisauniquea ne map from the interval inserted at a point x i to the interval inserted at any ( )x i = x j, this gives a well-defined means of extending to an action t of on the circle S t. As t! 0, this action approaches the original action in Hom(, Homeo(S 1 )), provided that one makes a reasonable identification between S 1 and the larger circles S t. Generalizing Denjoy s construction is the notion of semi-conjugacy, which we already hinted at in our discussion of equivalence of flat bundles in Section 2.1. Recall that two representations and 0 :! Homeo(S 1 ) are conjugate if there is a homeomorphism h of S 1 such that h ( )(x) = 0 ( )h(x), for all 2 and x 2 S 1. Semi-conjugacy is essentially the same notion, but h is no longer required to be a homeomorphism: it is permitted to collapse intervals to points, or have points of discontinuity where it performs the Denjoy trick, blowing up points into intervals. To make the definition completely formal requires a bit of care, so we defer the work to Section 5.1. There, we will also see that semi-conjugate representations have many of the same dynamical properties, so it is entirely reasonable to consider a semi-conjugacy to be a trivial deformation. As construction 3.4 can be applied to any action of a countable group, there is no hope that such groups could be locally rigid in the sense of Definition 3.1. Instead, we make the following obvious modification. 4 Often it is additionally required that the conjugacy be by an element of G close to the identity. 8

9 Rigidity Definition 3.5 (Local rigidity in Homeo(S 1 )). Let be a countable group. A representation :! Homeo(S 1 )islocally rigid if there is a neighborhood U of in Hom(,G) such that each 0 2 U is semi-conjugate to. Although Construction 3.4 produces an action by homeomorphisms, there is a way to construct semi-conjugate C 1 -actions of certain groups by a similar trick (due to Denjoy and Pixton). However, the construction cannot generally be made C 2, and the case of intermediate regularity is quite interesting! A detailed discussion of such regularity issues is given by Navas in [50, Chapt. 3]. 3.2 Full rigidity If one considers not only small deformations of a representation :! G, butarbitrary deformations, one is led to a statement about the connected component of in Hom(,G). Rigidity then should mean that the connected component is as small as possible. There is a particularly nice way to formalize this notion in the case where G = Homeo(S 1 ), we call this full rigidity. (Although global rigidity or strong rigidity would also make sense, these names were already taken!) Rigidity Definition 3.6 (Full rigidity). A representation :! Homeo(S 1 )isfully rigid if the connected component of Hom(, Homeo(S 1 )) containing consists of a single semi-conjugacy class. The set of all representations semi-conjugate to a given one is path-connected (see the remark after Definition 5.8), so this definition really does describe representations with the smallest possible connected component. An easy example of full rigidity is the following. We leave the proof as an exercise. Example 3.7. The inclusion of Z/mZ into Homeo(S 1 ) as a group of rotations is fully rigid. In fact, every action of any finite group on S 1 is fully rigid. A much less trivial example of fully rigid actions follows from a theorem of Matsumoto, Theorem 4.15, which we will state in Section 4.2. Example 3.8 (Consequence of Theorem 4.15). Let : g! Homeo(S 1 ) be a faithful representation with image a discrete subgroup of PSL(2, R). Then is fully rigid. In Section 4, we ll also describe a theorem on geometric actions subsuming both of the above examples (c.f. Theorem 4.17 geometric actions on S 1 are fully rigid ). 3.3 Strong (Mostow) Rigidity Although not our primary focus here, we mention briefly a third definition of rigidity coming from the tradition of lattices in Lie groups. A more in-depth introduction can be found in [34] or [56]. Rigidity Definition 3.9 (Strong rigidity). A lattice in a Lie group G is strongly rigid if any homomorphism :! G whose image is also a lattice extends to an isomorphism of G. Mostow proved strong rigidity for cocompact lattices in G = Isom(H n ), provided that n 3; this was extended by Prasad to lattices in isometry groups of other locally symmetric spaces. Margulis superrigidity theorem can be seen as a further generalization of this kind of result. But of interest to us is what happens to Mostow s original rigidity theorem when n = 2. The cocompact lattices in Isom(H 2 )=PSL(2, R) are surface groups, and they are not rigid in PSL(2, R). Rather, there is a 6g 6-dimensional Teichmüller space of PSL(2, R)-conjugacy classes of discrete, faithful representations g! PSL(2, R), forming two connected components of the 9

10 character variety Hom( g, PSL(2, R))/ PSL(2, R). As we mentioned earlier, these parametrize the (marked) hyperbolic structures on g. However, it is still possible to recover a kind of rigidity for these representations by thinking of PSL(2, R) as a group of homeomorphisms of the circle. Although not conjugate in PSL(2, R), any two discrete, faithful representations g! PSL(2, R) are conjugate by a (possibly orientationreversing) homeomorphism of S 1. To see this using a little hyperbolic geometry think of PSL(2, R) acting by Möbius transformations on the Poincaré disc model of H 2.Twodiscrete faithful representations of g will have homeomorphic fundamental domains, a homeomorphism between them can be extended equivariantly to a homeomorphism of the disc, including the boundary circle, which induces the conjugacy. We conclude this section by mentioning an even stronger rigidity result, in the spirit of di erentiable rigidity, that comes from considering the regularity of the conjugating homeomorphism. Theorem 3.10 ( Mostow rigidity on the circle [48]). Let and 0 be discrete, faithful representations of g into PSL(2, R); with h h 1 = 0. If the derivative h 0 (x) is nonzero at any point x where it is defined, then h 2 PSL(2, R). Further discussion can be found in the survey paper [1]. 3.4 The Zimmer program The strong rigidity theorems of Mostow, Margulis, and others place strict constraints on representations of lattices in Lie groups. The Zimmer program is a series of conjectures organized around the theme that actions of lattices on manifolds should be similarly constrained. (See Fisher s survey paper [20] for a nice introduction.) The main successes of the Zimmer program, such as Zimmer s cocycle rigidity, use ergodic theory and apply only to actions by measure-preserving homeomorphisms or di eomorphisms. However, there is also a small family of nice results regarding lattices acting on the circle. We give two examples; the first a beautiful instance of the Zimmer program catchphrase large (rank) groups do not act on small (dimensional) manifolds, the second a classification of all actions of lattices on S 1. Theorem 3.11 (Witte-Morris, [59]). Let be an arithmetic lattice in an algebraic semi-simple Lie group of Q-rank at least 2 (for example, a finite index subgroup of SL(n, Z), n 3). Then any representation :! Homeo(S 1 ) has finite image. Theorem 3.12 (Ghys, Theorem 3.1 in [25]). Let be an irreducible lattice in a semi-simple Lie group of real rank at least 2. (A basic example is G = PSL(2, R) PSL(2, R).) Up to semi-conjugacy, any action :! Homeo(S 1 ) either has finite image or is obtained by a projection of G onto a PSL(2, R) factor:,! G PSL(2, R) Homeo(S 1 ). Ghys theorem does not imply Witte-Morris result, as a representation that is semiconjugate to one with finite image need only have a finite orbit, not finite image. But the theorem does give new examples of fully rigid representations. Corollary Let be an irreducible lattice in PSL(2, R) G, whereg is a semi-simple Lie group with no PSL(2, R) factors. Then the representation : PSL(2, R) Homeo(S 1 )is fully rigid in Hom(, Homeo(S 1 )). Thompson s group. Thompson s group G is the group of piecewise-a ne homeomorphisms of the circle that preserve dyadic rational angles. Brown and Geoghegan and others have suggested thinking of it as the analog of a higher rank lattice subgroup for Homeo(S 1 ). It is also known to be rigid, at least if one restricts to C 2 actions. 10

11 Theorem 3.14 (Ghys Sergiescu [27]). All nontrivial homomorphisms semi-conjugate to the standard inclusion. : G! Di 2 (S 1 ) are To the best of our knowledge, it is an open question whether this result extends to rigidity of homomorphisms G! Homeo(S 1 ). 4 Two perspectives on rigidity of group actions Having seen some samples of rigidity of group actions, we now turn to a discussion of what phenomena (other than being a higher-rank lattice ) can lead to this rigidity. We focus on two perspectives that have particularly strong ties to actions of surface groups on the circle. The first, hyperbolicity, comes from classical smooth dynamics. Although hyperbolicity is an essentially C 1 notion it is a statement about derivatives we also present a slightly di erent notion of hyperbolicity for group actions due to Sullivan that can be applied to actions by homeomorphisms. As an illustration, we give two di erent proofs of structural stability for surface group actions, both with a hyperbolic flavor. Our second theme is geometricity. We motivate this with a discussion of maximal representations of surface groups into Lie groups and generalizations to Homeo(S 1 ). We argue that geometricity, rather than maximality, is the right way to think of the phenomenon underlying these rigidity results, and advertise rigidity of geometric representations as an organizing principle. 4.1 Rigidity from hyperbolicity Broadly speaking, hyperbolic dynamics deals with di erentiable dynamical systems that exhibit expanding and contracting behavior in tangent directions. The classic example of this is an Anosov di eomorphism. Definition 4.1 (Anosov di eomorphism). Adi eomorphism f of a manifold M is Anosov if there exists a continuous f-invariant splitting of the tangent bundle TM = E u E s, and constants > 1, C > 0 such that kdf n (v)k C n kvk for v 2 E u,n 0 and kdf n (v)k C n kvk for v 2 E s,n 0. A slight variation on this is an Anosov flow:aflowf t on M is Anosov if there is a continuous invariant splitting TM = E u E s E 0,whereE u and E s are expanding and contracting as in Definition 4.1, and E 0 is tangent to the flow. Hyperbolicity implies structural stability A major theme in hyperbolic dynamics is extracting rigidity from hyperbolic behavior. A basic instance of this is the following theorem of Anosov. Theorem 4.2 ([2]). Anosov flows are structurally stable. Theorem 4.2 has been generalized to many kinds of di eomorphisms and flows with weaker forms of hyperbolic behavior than the Anosov condition, such as partial hyperbolicity. This is still an active area of research the survey paper [11] provides a nice exposition of recent results in the partially hyperbolic case. We also recommend [35] for a general introduction to hyperbolic dynamics. Here we will focus on a very particular case of Anosov s original structural stability theorem, one that applies (of course!) to certain circle bundles over surfaces. Theorem 4.3 (Anosov [2]). Let g be a surface with hyperbolic metric, and E the unit tangent bundle of g. Then the geodesic flow on E is structurally stable. 11

12 The unit tangent bundle E has the structure of a flat circle bundle over g, its holonomy is the representation g! PSL(2, R) Homeo(S 1 ) defining the hyperbolic structure on the surface. What is more, the foliation transverse to the fibers also contains information about the geodesic flow: its tangent space is spanned by the flow direction E 0 together with the expanding direction E u. As a consequence, one can derive a structural stability result for these representations. Corollary 4.4 (Structural stability of group actions). Let : g! Di 1 (S 1 ) be faithful with image a discrete subgroup of PSL(2, R). Then is structurally stable in the sense of Definition 3.3. The proof is not di cult conceptually, but it requires that you know a little more about Anosov flows than we have introduced here. We give a quick sketch for the benefit of those readers who have the background. More details can be found in [23]. Proof sketch, following [23]. Let f t denote geodesic flow on the unit tangent bundle E of g. Let 0 : g! Di 1 (S 1 )beac 1 perturbation of, and E 0 the resulting flat bundle. This is equivalent to E as a smooth bundle, so we can think of it as the space E with a foliation F 0 transverse to the fibers. As 0 is C 1 -close to, the foliation F 0 is C 1 close to the transverse foliation for, whichistheweak unstable foliation E 0 E u of f t. As a consequence, the intersection of F 0 with the weak stable foliation of f t is a 1-dimensional foliation. This foliation can be parametrized by a flow g t that is C 1 -close to f t. Moreover, the weak unstable of g t is exactly F 0 although strong stable and unstable foliations are not invariant under reparametrization, the weak foliations are. Theorem 4.3 now implies that g t and f t are topologically conjugate. As such a topological conjugacy maps the weak unstable foliation of f t to that of g t, it defines a conjugacy of and 0. Ghys di erentiable rigidity. Building on Corollary 4.4, Ghys proved a remarkable di erentiable rigidity result for surface group actions. Theorem 4.5 (Local di erentiable rigidity [23]). Let : g! Di 1 (S 1 ) be smoothly conjugate to a discrete, faithful representation into PSL(2, R). Then any C 1 perturbation of is C 1 conjugate to a discrete, faithful representation into PSL(2, R). This is not quite local di erentiable rigidity (in the sense of Definition 3.2), as 0 is not necessarily C 1 conjugate to. However, in light of Theorem 3.10 Ghys result is the best one can hope for. Though quite technically involved, the idea of Ghys proof is to improve the topological conjugacy given by Corollary 4.4 to a smooth conjugacy by finding a 0 -invariant projective structure on S 1. Ghys later improved Theorem 4.5 to a global di erentiable rigidity result. Theorem 4.6 (Global di erentiable rigidity [24]). Let r 3, and let : g! Di r (S 1 ) be a representation with Euler number ± ( g ). Then is faithful, and conjugate by a C r di eomorphism to a representation into a discrete subgroup of PSL(2, R). The essential global ingredient is a theorem of Matsumoto on maximal representations, which we ll introduce in the next subsection. Matsumoto s theorem provides a topological conjugacy between and a representation into PSL(2, R); Ghys again finds an invariant projective structure that gives the C r conjugacy. Corollary 4.4 and Ghys local di erentiable rigidity theorem also apply to the representations to PSL (k) constructed in Example 2.10, as the k-fold fiberwise cover of the unit tangent bundle of g also admits an Anosov flow with weak-unstable foliation transverse to the fibers. In [5], Bowden uses this to prove a global di erentiable rigidity result. Theorem 4.7 ([5]). Let t : g! Di 1 (S 1 ) be a continuous path of representations, with 0 a discrete, faithful representation to PSL (k).then t is smoothly conjugate to 0. 12

13 Bowden s work produces a topological conjugacy between 0 and t, which by Ghys arguments in [24] must be a smooth conjugacy. Using our Theorem 4.17 instead of Bowden s results, one can make the same statement as Theorem 4.7 not just for deformations along paths, but for arbitrary representations in the same connected component of 0. Sullivan s Hyperbolicity implies structural stability. In [55], Sullivan gives a completely di erent definition of hyperbolic group action one that applies to actions by homeomorphisms on any compact metric space X. Recently, Kapovich, Leeb and Porti [38] have used this property in studying the boundary actions of discrete groups on symmetric spaces a situation where the best regularity at hand is really C 0, so traditional notions of hyperbolicity have nothing to say. We explain Sullivan s idea here, and use it to give an alternative proof of Corollary 4.4, without reference to flows, bundles, or di erentiability. Definition 4.8 (Expanding property [55]). A homomorphism f 2 Homeo(X) is expanding at x 2 X if there exists c>1 and a neighborhood U of x (called an expanding region) such that d(f(y),f(z)) > c d(y, z) for all y, z in U. An action :! Homeo(X) isexpanding if, for each x 2 X, there exists some ( ) that is expanding at x. Expanding actions allow one to code points of X by sequences in. Construction 4.9 (Coding by expanding regions). Suppose X is compact, and :! Homeo(X) is expanding. Let S be a finite generating set for, large enough so that the expanding regions for elements of S cover X. Acoding of x 0 2 X is a sequence s i 2 S, chosen so that x 1 := s 1 (x 0 ) lies in the expanding region for s 2, and inductively, x i := (s i )(x i 1 )liesin the expanding region for (s i+1 ). Using the expanding property, the point x 0 can be recovered from a coding sequence {s i }. If is chosen su ciently small (depending on the cover of X by expanding sets, and on the expanding constants for the s i ), and U(x n ) is any neighborhood of x n contained in the -ball about x n,then x 0 = \ (s n...s 1 s 0 ) 1 (U(x n )). n Sullivan s definition of a hyperbolic action is one where this coding is essentially unique. Definition 4.10 (Hyperbolicity à la Sullivan). An action is called hyperbolic if it is expanding, and the following condition holds: ( ) There exists N such that, if {s i } and {t i } are coding sequences for a point x, each initial segment s 1 s 2...s m has a counterpart segment t 1 t 2...t k such that (s 1...s m )(t 1...t k ) 1 can be represented by a word of length N in S. This brings us to Sullivan s structural stability theorem. Theorem Let be a finitely generated group, and X a manifold. Any hyperbolic 2 Hom(, Di 1 (X)) is structurally stable. More generally, if X is any compact metric space, :! Homeo(X) is hyperbolic, and 0 is a small perturbation of such that a set of expanding generators for remain expanding on the same covering sets, then Sullivan s proof of Theorem 4.11 shows that 0 is conjugate to. This happens, for example, if the generators of are perturbed by small bi-lipschitz homeomorphisms. The proof of Theorem 4.11 is remarkably simple. For each point x 2 X, one chooses a coding sequence {s i } and small balls B (x i ) about x i.defineh2homeo(x) by h(x) = \ n 0 (s 0 s 1...s n ) 1 (B (x n )). 13

14 The requirement that be expanding implies that h(x) is a singleton, and property ( ) implies that it is independent of the sequence {s i }. One then checks in a straightforward manner that h is continuous, injective, and defines a conjugacy. As any discrete, faithful representation : g! PSL(2, R) Homeo(S 1 ) is hyperbolic, we have the following corollary. Corollary 4.12 (Structural stability, [55].). Let : 1 (M)! Homeo(S 1 ) be faithful, with image a discrete subgroup of PSL(2, R). Then any su ciently small C 1 or small bi-lipschitz perturbation of is conjugate to. 4.2 Maximality and geometricity We turn now to our second theme maximality and geometricity, starting with a particular consequence of Theorem Theorem 4.13 (Goldman [29], [28]). If 2 Hom( g, PSL(2, R)) has (maximal) Euler number ± ( g ), then it is discrete and faithful and defines a hyperbolic structure on g. This theme that representations maximizing some characteristic number should have nice geometric properties has been extended to representations of surface groups into many other Lie groups. A semi-simple Lie group G is said to be of Hermitian type if it has finite center, no compact factors, and the associated symmetric space X has a G-invariant complex structure. This structure allows one to assign a characteristic Toledo number T ( ) to each representation : g! G. Analogous to the Euler class and Euler number for circle bundles, it satisfies an inequality T ( ) apple ( g )rank(x). (Proofs of special cases of this inequality were given by Turaev, and Domic and Toledo; the general case is due to Clerc and Ørsted.) Toledo showed that, in the special case G = SU(n, 1), representations maximizing this invariant have a nice geometric property they stabilize a complex geodesic, i.e. are conjugate into S(U(n 1) U(1, 1)). This result was gradually improved by a number of authors, culminating in the following theorem of Burger, Iozzi and Wienhard. Theorem 4.14 ( Maximal implies geometric [9]). Let G be a Hermitian Lie group. If : g! G has maximal Toledo invariant, then is injective and has discrete image. Moreover, there is a concrete structure theory that describes the image of. We have been intentionally vague about the structure theory, as it is somewhat complicated to state. A major ingredient in the proof is analyzing the Zariski closure of the image of g one views as a composition of two maximal representations g! H,! G where the image of g is Zariski dense in H, and then one studies separately the maximal representations g! H and the restrictions on the geometry of H as a subgroup of G. See[9], or for a gentler introduction, the survey papers [8] or [10]. Maximality in Homeo(S 1 ). Though Theorem 4.14 and its proof appear to rely heavily on the structures of Lie groups, there is a direct analog for representations of surface groups to Homeo(S 1 ), due to Matsumoto. Theorem 4.15 (Matsumoto s maximal implies geometric [45]). Let : g! Homeo(S 1 )be a representation with (maximal) Euler number ± ( g ). Then is faithful, and semi-conjugate to a representation with image a discrete subgroup of PSL(2, R). Even more surprising, there are a number of parallels between Matsumoto s proof and the proof of Theorem 4.14 in [9]. In [36], Iozzi applies tools developed to study representations of Lie groups (e.g. boundary maps and continuous bounded cohomology) to give a new proof of 14

15 Matsumoto s theorem, and a unified approach to studying maximal representations into SU(n, 1) and Homeo(S 1 ). Rigidity from geometricity As all discrete, faithful representations into PSL(2, R) are conjugate by homeomorphisms of the circle (c.f. Section 3.3), this gives the following full rigidity theorem. Corollary Representations in Hom( g, Homeo(S 1 )) with maximal Euler number are fully rigid. Equivalently, the sets of representations with Euler number ( g ) and with Euler number ( g ) each consist of a single semi-conjugacy class. It is our contention that the phenomenon underlying this rigidity is geometricity Matsumoto s theorem states that maximal representations are geometric in the sense of Definition 2.8. The following theorem was proved in the case of surface groups in [42]. Theorem 4.17 ( Geometricity implies rigidity [42]). Let : representation. Then is fully rigid.! Homeo(S 1 ) be a geometric This theorem not only implies Corollary 4.16, but also gives rigidity for the representations of surface groups constructed in Example 2.10, and rigidity of representations of other lattices in PSL (k). We also conjecture that geometricity is the only source of full rigidity for actions of surface groups. Conjecture Suppose that : g! Homeo(S 1 ) is fully rigid. Then is geometric. In the next two sections (5 and 6) we introduce the tools needed to prove Theorem 4.17 which are useful and beautiful in their own right and discuss a number of other applications of these tools. An outline of the proof of Theorem 4.17 is given in Section 7, and a hint at the conjecture in Section Not just surface groups From surfaces, one can build many other interesting actions of groups on the circle. A particularly nice example is the most basic case of the universal circle action, due to Thurston. Suppose that is the fundamental group of a hyperbolic 3-manifold that fibers over the circle, with fiber g.then has a presentation = h g,t t t 1 = ( )i for some 2 Out( g ). If : g! PSL(2, R) is a representation with Euler number 2g + 2, then it can be extended in a natural way to a representation of. To see this, we use the hyperbolic structure on g given by. Let f be a di eomorphism of g inducing on g (such a di eomorphism exists since g is a K(, 1) space), and lift f to a di eomorphism ˆf of the universal cover H 2 of g that is equivariant with respect to the action of g on H 2. The di eomorphism ˆf admits a unique extension to a homeomorphism of the circle at 2 = S 1, and we define (t) to be this homeomorphism. That f is compatible with the action of g on H 2 implies that satisfies the group relations (t t 1 )= ( ( )), so is well defined. Moreover, this extension of g to a representation of is unique. One way to see this is by considering fixed points of elements ( ) for 2 g. The relation t t 1 = ( ) implies that (t) necessarily sends the (unique) attracting fixed point of ( ) to the attracting fixed point of ( ). Attracting fixed points of elements of ( ) PSL(2, R) are dense in S 1, so this property determines (t). Since was maximal on g, it is fully rigid, and therefore this unique extension is also rigid. For readers familiar with the Thurston norm on homology of a 3-manifold (c.f. [57]), the construction above can be summarized as follows. 15

16 Theorem Let M be a hyperbolic 3-manifold that fibers over the circle, and let = 1 (M). For each fibered face F of the Thurston norm ball of H 2 (M; R), there is a representation F :! Homeo + (S 1 ) that is fully rigid in Hom(, Homeo(S 1 )). Furthermore, Ghys di erentiable rigidity (Theorem 4.6) together with Theorem 3.10 implies that the component of Hom(, Homeo + (S 1 )) containing F contains no representation with image in Di 3 (S 1 ). A typical fibered 3-manifold M fibers over the circle in many di erent ways, giving di erent, non semi-conjugate, fully rigid actions of 1 (M) on S 1. In the language above, if F and F 0 are two di erent fibered faces of the Thurston norm ball, then F and F 0 are not semi-conjugate. The construction has been generalized to give actions of fundamental groups of other 3- manifolds. Instead of a fibration of M over S 1, one can use a pseudo-anosov flow on M ([3], [19]), a quasi-geodesic flow [13], or even certain types of essential lamination [14], to get analogous faithful actions of 1 (M) on S 1. It would be interesting to investigate rigidity of these examples using the techniques described below. 5 Rotation number theory One approach to understanding the topology of Hom(,G) is to look for natural coordinates on the space this approach has been very fruitful when G is a linear group. The perspective we promote here, originally due to Calegari (see e.g. [15]), is that the representation spaces Hom(, Homeo(S 1 )) also have natural coordinates, given by rotation numbers. To motivate our discussion, we begin with a few words on the linear case. Motivation: trace coordinates on character varieties. It is a well-known fact that linear representations of a group over a field of characteristic zero are isomorphic if and only if they have the same characters. In many situations, this fact can be turned into a system of coordinates on the space of representations of. Consider G = SL(2, C) as a concrete example. When is a finitely generated group, irreducible representations! SL(2, C) are determined up to conjugacy by the traces of finitely many elements; so a finite collection of traces can be used to parametrize Hom(, SL(2, C))/ SL(2, C), at least away from the reducible points. The more sophisticated algebraic-geometric quotient of Hom(, SL(2, C)) hinted at in Section 2.3 that gives the SL(2, C) character variety for collapses reducible representations of the same character to points, and so traces really do give coordinates on this variety. This perspective was originally promoted by Culler and Shalen [17], who gave an algebraic compactification of components of the SL(2, C) character variety when is a hyperbolic 3-manifold group. A more recent, and perhaps more direct application of trace coordinates can be found in [31]. Here coordinates are used to study the action of the mapping class group of on Hom( 1 ( ), SL(2, C))/ SL(2, C) in the case where is a once punctured torus. This paper also contains many pictures of slices of the character variety (at least the real points!) drawn in trace coordinates. Characters for Homeo(S 1 ). Motivated by the linear case, in order to parametrize representations into Homeo(S 1 ) up to conjugacy we should look for a class function on Homeo(S 1 )toplay the role of the character. As trace also captures significant dynamical information (e.g. the trace of an element of PSL(2, C) determines the translation length of its action on hyperbolic 3-space), we would hope to find a class function on Homeo(S 1 ) that holds some dynamical information as well. Fortunately, such a function exists: this is the rotation number of Poincaré. 5.1 Poincaré s rotation number Let Homeo Z (R) denote the group of orientation-preserving homeomorphisms of R that commute with integer translations. Homeo Z (R) can be thought of as the group of all lifts of orientationpreserving homeomorphisms of S 1 = R/Z to its universal cover R, and fits into a central 16

17 extension 0! Z! Homeo Z (R)! Homeo(S 1 )! 1 where the inclusion of Z is as the group of integer translations. Notational remark. It is standard to use the notation g for a lift of g 2 Homeo(S 1 )to Homeo Z (R). However, occasionally especially in later sections we will want to emphasize the perspective that Homeo Z (R) should be considered the primary object, and Homeo(S 1 ) its quotient. In this case will use g to refer to an element of Homeo Z (R) and ḡ its image in Homeo(S 1 ). Definition 5.1 (Poincaré). Let g 2 Homeo Z (R) and x 2 R. The(lifted) rotation number rot( g) is given by g n (x) rot( g) := lim n!1 n. For g 2 Homeo(S 1 ), the R/Z-valued rotation number of g is defined by rot(g) := rot( g) mod Z where g is any lift of g to Homeo Z (R). It is a classical result that these limits exist and are independent of choice of point x. An elegant and elementary proof is given in [52] the key is the observation that g 2 Homeo Z (R) implies that g n (x) x ( g n (y) y) apple 1 (1) and that this both implies that the sequence gn (x) n is Cauchy, hence converges, and that its limit is independent of x. In fact, this proof applies not only to elements of Homeo Z (R) buttoawider class of maps. Definition 5.2 (Monotone maps). A monotone map of R is a (not necessarily continuous) map f : R! R satisfying i) Z-periodicity: f(x + 1) = x + 1 for all x 2 R and ii) monotonicity: x apple y () f(x) apple f(y), for all x, y 2 R. Perhaps these would be better called Z-periodic monotone maps so as to emphasize both conditions. We hope the reader will forgive us for choosing the shorter name, and keep condition i) in mind. As advertised, rotation number is a class function that captures a wealth of dynamical information. The following properties (perhaps with the exception of continuity) are relatively easy to check just using the definition. Proposition 5.3 (First properties of rot). Let g 2 Homeo Z (R). 1. Let T denote translation by. Then rot(t ) =. 2. (Homogeneity) rot(g n )=n rot(g). 3. (Conjugacy invariance) rot(hgh 1 ) = rot(g) for any h 2 Homeo Z (R) 4. (Quasi-additivity) rot(f g) rot(f) rot(g) apple 2 5. (Continuity) rot is a continuous function. 6. (Periodic points) rot(g) = 0 if and only if g has a fixed point. More generally, rot(g) = p/q 2 Q if and only if there exists x such that g q (x) =x + p, in this case the projection of g to Homeo(S 1 ) has a periodic orbit of period q. These statements also hold for rot(g) wheng 2 Homeo(S 1 ) and 1-3 are easily seen to hold when g is a monotone map. 17

18 / As an example of using the definition, we give a quick proof of quasi-additivity 5. Proof of quasi-additivity. By composing with integer translations, which commute with f and g and do not a ect the inequality, we may assume without loss of generality that f(0) 2 [0, 1) and g(0) 2 [0, 1). This gives 0 apple fg(0) < 2. Also, using x = 0 in the definition of rot gives the estimates rot(f) 2 [0, 1], rot(g) 2 [0, 1], and rot(fg) 2 [0, 2], from which the desired inequality follows. Although less obvious than the statement about periodic points in Proposition 5.3, we can still extract dynamical information from the rotation number when rot(g) /2 Q. For this, we need to (as promised, finally!) properly introduce the notion of semi-conjugacy. Semi-conjugacy will play a central role in the rest of this paper in fact, we hope to convince the reader that it is the right notion of equivalence for actions by homeomorphisms on the circle. 5.2 Semi-conjugacy In section 3, we loosely defined a semi-conjugacy to be a kind of conjugacy with Denjoy-ing allowed. We now make this idea precise, eventually giving two equivalent definitions. The first, due to Ghys, comes from the beautiful insight that things are much simpler when lifted to R. Definition 5.4 (Semi-conjugacy à la Ghys). Let be any group. Two representations 1, 2 :! Homeo Z (R) are semi-conjugate if there exists a monotone map h : R! R such that h 1 ( )(x) = 2 ( )h(x) for all x 2 R and 2 The map h is called a semi-conjugacy between 1 and 2. Building on this, we say that representations 1 and 2 :! Homeo(S 1 ) are semi-conjugate if there exists a central extension ˆ of, and semi-conjugate representations ˆ 1 and ˆ 2 : ˆ! Homeo Z (R) such that the following diagrams commute for i =1, 2 0 / Z / ˆ / 1 ˆ i i 0 / Z / Homeo Z (R) / Homeo(S 1 ) / 1 Note that ˆ is uniquely defined, it is the pullback of the central extension Z! Homeo Z (R)! Homeo(S 1 )by 1 (and also the pullback by 2 the definition of semi-conjugacy implies in particular that these pullbacks are isomorphic as extensions of ). Remark 5.5 (A word of warning). Definition 5.4 appears in [21], but with inconsistent use of pullbacks to Homeo Z (R), an unfortunate little mistake which has caused quite a bit of confusion! To see what goes wrong when one fails to pull back, take h : R! R to be the floor function h(x) =bxc. This is a monotone map, and it descends to the constant map h(x) = 0 on S 1 = R/Z. Now any representation :! Homeo(S 1 ) appears to be semi-conjugate 6 to the trivial representation via h, as 0= h ( )(x) = h(x) for all x 2 S 1 However, if we instead take a (pullback) representation ˆ to Homeo Z (R), the situation is di erent. Suppose that ˆ is semi-conjugate to a pullback ˆ triv of the trivial representation via h. Up to 5 With a more involved argument, it is possible to reduce the bound from 2 to 1 we ll see this as a consequence of Theorem 6.16 below 6 of course, not with the correct definition of semi-conjugacy! 18

19 multiplying by an element of the center of ˆ, each 2 ˆ satisfies ˆ triv ( )=id. In this case, we have bˆ ( )(x)c = bxc for all x 2 R. In particular, taking x 2 Z gives that ˆ ( ) ±1 (x) x, which implies that x is fixed by ˆ ( ). Thus, ( )fixes02s 1. With some work, this argument can be improved to show generally that a representation :! Homeo(S 1 ) is semi-conjugate to the trivial representation if and only if it has a global fixed point. One of the benefits of lifting to the line is that it makes it easy to show semi-conjugacy is an equivalence relation. Proposition 5.6. Semi-conjugacy is an equivalence relation Proof. (Following [21, Prop 2.1]). It su ces to prove that semi-conjugacy is an equivalence relation on representations to Homeo Z (R), since in the case of representations to S 1, the extension ˆ is uniquely defined. For representations to Homeo Z (R), reflexivity of semi-conjugacy is immediate, and transitivity an easy exercise. For symmetry, suppose that and 0 :! Homeo Z (R) satisfy ( )h = h 0 ( ) for some h : R! R as in Definition 5.4. Define h 0 : R! R by h 0 (x) =sup{y 2 R h(y) apple x}. Then h 0 (x+1) = sup{y 2 R h(y 1) apple x} = h 0 (x)+1; and x 1 apple x 2 implies that h 0 (x 1 ) apple h 0 (x 2 ), so h is monotone. Moreover, by construction we have h 0 ( )(x) =sup{y 2 R h(y) apple ( )(x)} =sup{y = 0 ( )(z) 2 R h(z) apple x}, since h is order preserving = 0 ( )h 0 (x) so h 0 is a semi-conjugacy between 0 and. Semi-conjugacy and minimal actions. The next definition of semi-conjugacy is due to Calegari and Dunfield, originally appearing in [14, Def. 6.5, Lem. 6.6]. It avoids lifting to R by describing semi-conjugate representations as those having a kind of common reduction. Definition 5.7 (Degree one monotone map). A map h : S 1! S 1 is called a degree one monotone map if it admits a continuous lift h : R! R that is monotone as in Definition 5.2. Definition 5.8 (Semi-conjugacy à la Calegari Dunfield). Two representations 1 and 2 in Hom(, Homeo + (S 1 )) are semi-conjugate if and only if there is a third representation :! Homeo + (S 1 ) and degree one monotone maps h 1 and h 2 of S 1 such that i h i = h i. Moreover, if 1 has a finite orbit, then can be taken to be conjugate to an action by rotations and h 1 a (discontinuous) map with image the finite orbit of 1. In this case, 2 necessarily has a finite orbit as well. Otherwise, can be taken to be a minimal action, i.e. with all orbits dense; and h i a map collapsing each wandering interval for the action of i to a point. (See [26, Prop 5.6] for more justification of this description). In this way, one thinks of as capturing the essential dynamical information of i. This description also emphasizes the fact that any semi-conjugacy between two minimal actions is a genuine conjugacy, and it can be used to show that semi-conjugacy classes are connected. 19

20 Rotation numbers detect semi-conjugacy. Now we may finally complete our list of properties of the rotation number. The following proposition is essentially a theorem of Poincaré, a proof and further discussion can be found in [26, Sect. 5.1]. Proposition 5.3, continued (Further properties of rot). 5. For g 2 Homeo(S 1 ), rot(g) is irrational if and only if g is semi-conjugate to an irrational rotation. This semi-conjugacy is actually a conjugacy if g has dense orbits. 6. (Detects semi-conjugacy) More generally, f and g in Homeo Z (R) are semi-conjugate if and only if rot(f) = rot(g). Property 6 says that rotation number is a complete invariant of semi-conjugacy classes in Homeo Z (R) or Homeo(S 1 ). In other words, an action of Z is determined up to semi-conjugacy by the rotation number of a generator. In Section 5.4 we will see how to generalize this to actions of other groups, using rotation numbers as coordinates on the quotient of Hom(, Homeo(S 1 )) by semi-conjugacy. But first, we specialize again to surface groups, and return to the definition of the Euler number. 5.3 The Euler number as a rotation number The goal of this section is to give two concrete descriptions of the Euler number of a circle bundle over a surface. The first, using rotation numbers, is specific to flat bundles and one of the main ingredients in Milnor s (and Wood s) original proofs of the Milnor Wood inequality. The second definition is classical and obstruction-theoretic; our description is intended to make it easy to reconcile the two. Consider the standard presentation of a surface group g = ha 1,b 1,...a g,b g Q i [a i,b i ]i. We will exploit the fact that this group has a single relator, made up of commutators, to assign an integer to each representation : g! Homeo(S 1 ). The reason commutators are useful is that they have cannonical lifts to Homeo Z (R) for any f and g 2 Homeo(S 1 )withlifts g and f 2 Homeo Z (R), the homeomorphism [ f, g] is independent of the choice of lifts. We ll use the notation rot[f,g] := rot([ f, g]), and similarly rot ([f,g][h, k]) := rot([ f, g][ h, k]) etc. for products. The following definition is implicit in [47] and used explicitly in [60]. Definition 5.9. The Euler number of a representation : g! Homeo(S 1 ) is the integer! gy e( ) := rot [ (a i ), (b i )]. i=1 Since rot is continuous, it follows immediately that e is continuous on Hom( g, Homeo(S 1 )). Of course, this definition only makes sense for flat bundles. Construction 5.10 below gives a definition applicable to arbitrary circle bundles over surfaces. (See also Definition 2.7). Construction 5.10 (Euler number as an obstruction class). Build g by taking a single 0-cell x, attaching a loop to x for each generator, and then gluing the boundary of a 2-cell to the loop based at x corresponding to the word Q [a i,b i ]. Cutting this complex along the 1-skeleton gives the familiar picture of g as a 4g-gon with sides identified. Given an S 1 bundle E over g, there is no obstruction to building a section over the 0-cell, or extending this to a section over the 1-skeleton since S 1 is connected. However, there is an obstruction to extending the section over the 2-cell. The easiest way to describe this obstruction is to pull back the bundle to an S 1 bundle over the universal cover g! g. Since g is contractible, the pull-back bundle trivializes as g S 1. A section over the 1-skeleton of E corresponds to a section of g S 1 over the 1-skeleton that is equivariant with respect to the action of 1 ( g ). The restriction of such a section to the boundary of a 2-cell in g gives a map S 1! g S 1, and projection to the S 1 factor a map S 1! S 1. The winding number of this 20

21 map is the only obstruction to extending it over the 2-cell, and is exactly the Euler number of the bundle. Now our description of a flat bundle as a quotient ( g S 1 )/ 1 ( g ) should make it possible to reconcile Construction 5.10 with Definition 5.9. We leave this as an exercise for the reader. The truly enthusiastic can also try to reconcile Construction 5.10 with Definition 2.7 (interpreted the right way, the construction gives a means of assigning integers to 2-cells, so is a 2-cocycle...) 5.4 Rotation numbers as coordinates In [21], Ghys gave a cohomological condition on representations in Hom(, Homeo(S 1 )) equivalent to the representations being semi-conjugate 7. Matsumoto later translated this into a statement about rotation numbers. Before stating this condition, note that for any two elements f, g 2 Homeo + (S 1 )withlifts f and g in Homeo Z (R), the number does not depend on the choice of lifts f and g. (f,g) := rot( f g) rot( f) rot( g) Theorem 5.11 ( Rotation numbers are coordinates [21], [44]). Let be a group with generating set S. Two representations and 0 in Hom(, Homeo(S 1 )) are semi-conjugate if and only if the following two conditions hold i) rot( (s)) = rot( 0 (s)) for each generator s 2 S. ii) ( ( 1 ), ( 2 )) = ( 0 ( 1 ), 0 ( 2 )) for each pair of elements 1 and 2 in. Matsumoto s proof consists in showing that the conditions i) and ii) given above imply that and 0 satisfy the cohomological condition given by Ghys 8. We give a self-contained proof in the spirit of Ghys, but without reference to cohomology. However, the reader familiar with H 2 b should be able to find it lurking in the background. Proof of Theorem Suppose that and 0 satisfy the conditions of Theorem For each generator s of,pickalift s of (s) and s 0 of (s 0 ) to Homeo Z (R) such that rot( s) = rot( s 0 ), this is possible by condition i). Note that condition ii) implies (inductively) that for any string of generators s 1 s 2...s n we have rot( s 1 s 2... s n ) = rot( s 10 s s 0 n ). (2) Let ˆ denote the pullback (Homeo Z (R)). By definition, ˆ is the subgroup of Homeo Z (R) generated by ( s, s) and (T,id) wheret (x) =x+1; and ˆ : ˆ! Homeo Z (R) is just projection onto the Homeo Z (R) factor. Equation (2) implies that the homomorphism : ˆ! 0 (Homeo Z (R)), given by ( s, s) =( s 0,s 0 ), for s 2 S (T,id) = (T,id) is an isomorphism of central extensions if some word (T k s i1... s in,s i1...s in ) in the generators of ˆ is trivial, then s i1...s in = id 2,so s i1... s in is an integer translation, and (2) implies that s 0 i 1... s 0 i n is the same translation. Hence (T k s 0 i 1... s 0 i n,s i1...s in ) is trivial as well. 7 Ghys condition is that the representations determine the same bounded integer Euler class in Hb 2 ( ; Z), we ll say a bit more about this in Section The key observation is that is a cocycle representative for the bounded real Euler class. 21

22 Moreover, the representation ˆ 0 : ˆ! Homeo Z (R) definedbyˆ 0 ( s, s) = s 0 gives a commutative diagram 0 / Z / ˆ / 0 / 1 ˆ / Z / Homeo Z (R) / Homeo(S 1 ) / 1 It remains to construct a semi-conjugacy between ˆ and ˆ 0. For x 2 R, define h(x) :=sup 2ˆ ˆ ( ) 1 ˆ 0 ( )(x). To see that this supremum is always finite, note that (2) implies that rot(ˆ ( )) = rot(ˆ 0 ( )), so quasi-additivity applied to ˆ ( ) 1 and ˆ 0 ( ) gives the bound rot(ˆ ( ) 1 ˆ 0 ( )) apple 2. The definition of rot now implies that x 3 apple ˆ ( 1 )ˆ 0 ( )(x) apple x + 3 and this holds for any 2 ˆ. As in the proof of Proposition 5.6, it is also easy to check that h is a monotone map, and so we just need to verify that h defines a semi-conjugacy. Let 2 ˆ, then hˆ 0 ( )(x) =sup 2ˆ =sup 2ˆ =sup 2ˆ which is exactly what we needed to show. =ˆ ( )sup 2ˆ =ˆ ( )h(x) ˆ ( ) 1 ˆ 0 ( )ˆ 0 ( )(x) ˆ ( ) 1 ˆ 0 ( )(x) (ˆ ( 1 ) 1 ˆ 0 ( )(x) ˆ ( ) 1 ˆ 0 ( )(x) Remark Theorem 5.11 has an alternative formulation and proof that also applies to semigroups, due to Golenishcheva-Kutuzova, Gorodetski, Kleptsyn, and Volk [32]. Since any semi-conjugacy between minimal actions is a genuine conjugacy, another way to put Theorem 5.11 is to say that the conjugacy class of a minimal action is determined by rotation numbers of generators and the function. In[32] it is shown that, if both the (positive) semigroup generated by a collection s i 2 Homeo + (S 1 ), and the (negative) semigroup generated by the {s 1 i } act minimally, then the collection of s i is determined up to conjugacy by their rotation numbers and the restriction of to the semigroup generated by the s i. (It would be nice not to have to consider the negative semigroup at all, but this hypothesis really is necessary see Example 1 in [32].) 5.5 Rotation number as a quasimorphism Perhaps it sounds obvious when stated this way, but the reason that rot can capture so much information about homeomorphisms of S 1 is because it is not a homomorphism. (In fact, Homeo(S 1 ) is a simple group, so any homomorphism to R/Z is necessarily trivial.) Here is an instructive example of how additivity can fail: Example Let f(x) =bxc and g(x) =bx +1/2c 1/2. Then f and g are monotone maps and each has a fixed point, so rot(f) = rot(g) = 0. However fg(0) = 1 and hence rot(fg)= 1. One can modify this example so that f and g are not just monotone maps, but lie in Homeo Z (R) our computation only used the property that f and g both had fixed points, but fg(x) =x 1 for some x 2 R. In fact, one can find such f and g that are lifts of hyperbolic elements in PSL(2, R) Homeo(S 1 ). 22

23 But there is one particular situation where rot is additive; we include it here, as it gives a nice reminder of the definition of rot and because we ll need to use the result later. Lemma Let f and g 2 Homeo Z (R) satisfy fg = T k where T k denotes the translation T k (x) :=x + k, for k 2 Z. Then rot(f) + rot(g) =k. Proof. That fg = T k implies that f = T k g 1,hence rot(f) = rot(t k g 1 T kn g n (x) g n (x)+kn )= lim = lim = rot(g 1 )+k = rot(g)+k. n!1 n n!1 n Despite not being a homomorphism, rot is a quasimorphism. A function r from a group G to R is called a quasimorphism if there exists D 0 such that r(fg) r(f) r(g) <Dfor all f,g 2 G. You can think of this as saying that r is a homomorphism up to bounded error. The optimal bound D is called the defect of r. The existence of a bound D for the function rot is exactly the quasi-additivity property of Proposition 5.3. In fact, the following is true. Proposition 5.15 (Prop 5.4 in [26]). Rotation number is the unique quasimorphism from Homeo Z (R) tor that is a homomorphism when restricted to one generator groups and that takes the value 1 on translation by 1. While finding the defect of rot is not too di cult (spoiler: it s D = 1), it is a di erent question to find the optimal bound for rot(f g) rot(f) rot(g) if f and g have specified rotation numbers. Problem Let f and g 2 Homeo Z (R). Given rot(f) and rot(g), what are the possible values of rot(fg)? More generally, if w is a word in f and g, what are the possible values of rot(w)? This was for many years an open problem, with significant implications in topology. This connection to topology was through the classification of taut foliations on Seifert fibered 3- manifolds. (This should, perhaps, not strike the reader as not so surprising, given the relationship between foliations and group actions discussed in Section 2.1.) Specifically, in [18], Eisenbud, Hirsch and Neumann reduced the last step of the program to classify such foliations to solving Problem 5.16 for the word fg. A conjectural answer to this question, and quite a bit of evidence, was later given in [37], and the problem finally solved by Naimi [49], more than 10 years after Eisenbud, Hirsch and Neumann s work. More recently, Calegari and Walker [15] developed an algorithm which answers Problem 5.16 not only for fg but for any word in f and g. This algorithm is elegant and elementary, implementable by computer or by hand, and the main tool used to prove our rigidity Theorem Quasimorphisms and H 2 b (G; R) Before describing the algorithm of Calegari Walker and the solution to Problem 5.16, we make a quick comment on the connections between quasimorphisms and bounded cohomology. Section 6 of [26] provides a wonderful introduction to the subject and its applications to groups acting on the circle; we see no need to compete with it here so will be very brief in our exposition. Bounded cohomology. One construction of the cohomology H (G; R) of a discrete group is as the cohomology of the complex C n (G; R) of functions G n! R, equipped with a suitable di erential. Bounded cohomology is the cohomology of the subcomplex Cb n (G; R) of bounded 23

24 functions. The resulting theory is remarkably di erent. For example, amenable groups always have trivial bounded cohomology in all degrees, while hyperbolic groups (including our favorite g) have infinite dimensional second bounded cohomology. A more sophisticated way to give the definition of a quasimorphism on a group G is as a inhomogeneous real 1-cochain on G, whose coboundary is bounded. The coboundary of a quasimorphism r : G! R is thus a bounded 2-cocycle on G, representing an element of second bounded cohomology, Hb 2 (G; R). In the case of the quasimorphism rot, the boundary 2-cocycle is exactly the function from Theorem Many other results on rotation numbers of group actions can also be rephrased in terms of bounded cohomology. For instance, Proposition 5.15 can be interpreted as a statement that Hb 2(Homeo(S1 ); R) is one dimensional, and Ghys original statement of Theorem 5.11 is a statement about the bounded integer Euler class in Hb 2( g; Z). See [26] and [21] for more details. 6 Rotation numbers of products In this section we explain the algorithm of Calegari and Walker and its first applications, including a solution to Problem 5.16 and a short proof of the Milnor Wood inequality. As a starting point, we return to the basic question raised in Section 5.5. Given rot(f) and rot(g), andawordw in f and g, what are the possible values of rot(w)? We will consider an even more general version of this problem Given constraints rot(f 1 )=s 1,..., rot(f n )=s n,andawordw in f 1,f 2,...,f n,what are the possible values of rot(w)? Our first lemma reduces this to a much easier problem. The lemma like much of the material in this section originally appears in [15]. Lemma 6.1. Let w be any word in f 1,...,f n.theset{ rot(w) rot(f i )=s i } is bounded and connected, hence an interval. Proof. In the n = 2 case, boundedness follows immediately from the fact that rot is a quasimorphism. The general case is proved by an inductive argument. For connectedness, property 6 of Proposition 5.3 states that any set of the form {f 2 Homeo Z (R) rot(f) =s} is a single semi-conjugacy class, hence connected. Since rot is continuous, { rot(w) rot(f i )=s i } is connected. Thus, to determine all possible values of rot(w(f 1,...,f n )) given the constraints rot(f i )=s i, it su ces to determine the maximum and minimum possible values. To this end, let R w (s 1,...,s n ):=sup{ rot(w) rot(f i )=s i }. We leave it to the reader to check that inf{ rot(w) rot(f i )=s i } = R w ( s 1,..., s n ), so a study of R w (s 1,...,s n )su ces. In [15], a compactness argument is used to show that the supremum is actually a maximum, although we won t need this fact here. Using continuity of rot, they also show the following. Proposition 6.2 (see Lemmas 2.14 and 3.3 in [15]). R w satisfies the following properties 1. (Lower semi-continuity) Let s i (k) be a sequence approaching s i.then R w (s 1,...,s n ) lim sup R w (s 1 (k),...,s n (k)). 2. (Monotonicity) If s i s 0 i,then R w (s 1,...,s n ) R w (s 0 1,...,s 0 n). 24

25 In fact, lower semi-continuity and the argument that { rot(w) rot(f i )=s i } is an interval apply not only to words in the letters f 1,...,f n, but to words that include their inverses f1 1,...f n 1 also. However, monotonicity fails in the case where inverses are permitted. Going forward, we will make frequent use of monotonicity, and so never permit inverses. Calegari Walker emphasize this by referring to a word in n letters as a positive word, wewill keep our convention and just say word or word in n letters. An nice consequence of Proposition 6.2 is the following. Corollary 6.3. Let w be a word in f 1,...,f n.thenr w (s 1,...,s n ) is completely determined by its values for rational s i. Proof. This is just the standard statement that an increasing, lower semi-continuous function is determined by its values on a dense set. 6.1 The Calegari Walker algorithm The Calegari Walker algorithm is a process to compute R w (s 1,...,s n ) for rational s i. It takes as input a possible cyclic order of a set of fixed or periodic points for homeomorphisms f i, and as output gives the maximum possible value of rot(w(f 1,...,f n )) where f i are homeomorphisms subject to these constraints. By running the algorithm on all of the (finitely many) possible cyclic orderings of periodic orbits with a given period s i, and taking the maximum of the outputs, one recovers R w. The description of the algorithm in [15] is designed to be easily implementable by computer. We have a di erent aim here, so will describe it in slightly di erent language. The main idea is remarkably simple one simply replaces homeomorphisms with monotone maps to reduce the computation to a finite problem. To explain this precisely requires a small amount of set-up, which we do now. Definition 6.4 ( Periodic elements of Homeo Z (R)). An element f 2 Homeo Z (R)isp/q periodic if rot(f) =p/q 2 Q. In other words, periodic elements of Homeo Z (R) are lifts of homeomorphisms of S 1 with periodic points. Definition 6.5 (Periodic orbits in R). Let f 2 Homeo Z (R) bep/q periodic. If p/q is in lowest terms 9,wedefineap/q-periodic orbit for f to be any orbit X that projects to a periodic orbit for f 2 Homeo(S 1 ). If instead p/q = kp 0 /kq 0 where p 0 /q 0 is in lowest terms, then a p/q-periodic orbit is defined to be a union of k disjoint p 0 /q 0 -periodic orbits for f. Much like the definition of rotation number, Definitions 6.4 and 6.5 naturally extend to all monotone maps of R. The next definition describes periodic monotone maps that translate points as far as possible to the right, given a constraint on an orbit. Definition 6.6 (Maximal monotone map). Let p/q 2 Q, withq>0. Let X R be a lift of a set X S 1 with X = q, enumerated in increasing order as... < x i 1 <x i <x i+1 <... The (X, p/q) maximal monotone map is the map f : R! R defined by f(x) =x i+p for x 2 (x i 1,x i ]. We ll call a set X R as described in Definition 6.6 a Z-periodic set of cardinality q. Note that the enumeration of X satisfies x i+q = x i + 1, and that the (X, p/q) maximal monotone map satisfies 9 Our convention is that in lowest terms implies that q>0. We also say 0/1 is the expression of 0 in lowest terms. 25

26 g f g f x 1 y 1 x 2 y 2 x 3... x 7 Figure 1: A periodic orbit for fg (Example 6.10) i. (Z-periodicity) f(x + 1) = f(x) + 1, so it really is a monotone map, and ii. (p/q-periodic orbits) f q (x i )=x i+pq = x i + p. The following lemma explains the terminology maximal, its proof is immediate from the definition. Lemma 6.7. Let X be a Z periodic set of cardinality q, and let f be the (X, p/q) maximal monotone map. If g is any p/q periodic monotone map with X as a p/q-periodic orbit, then As a consequence, we have the following. g(x) apple f(x) for all x 2 R. Proposition 6.8. Let p i /q i be given, and let f i be (X i,p i /q i ) maximal monotone maps. If g i are any other (X i,p i /q i ) monotone maps, and w a word in n letters, then rot(w(g 1,...g n )) apple rot(w(f 1,...f n )). Proof. Lemma 6.7 implies that g i (x) apple f i (x) for all x 2 R, hencew(g 1,...g n )(x) apple w(f 1,...f n )(x), and so rot(w(g 1,...g n )) apple rot(w(f 1,...f n )). Proposition 6.9. Let f i be (X i,p i /q i ) maximal monotone maps, and let w be a word in the f i. Then rot(w) 2 Q. Proof. Let X := S i X i, and note that X is invariant under integer translations, and that w(x) X. SinceX mod Z is finite, it follows that it contains some periodic point for w, i.e.a point x such that w m (x) =x + n for some m, n 2 Z. From the definition of rotation number, we have rot(w) =m/n. Proposition 6.9 also gives an easy and e ective way to compute rot(w) given (X i,p i /q i ). Starting at any x 2 X and successively applying w, one will eventually find k, m such that w k+m (x) =w k (x)+n, and hence can conclude that rot(w) =n/m. Here is a concrete example. Example Let p 1 /q 1 = p 2 /q 2 =1/2. Let X and Y R be Z-periodic sets of cardinality 2, and assume they are ordered...x 1 <y 1 <x 2 <y 2 <x 3... as shown in figure 1. Let f and g be (X, p 1 /q 1 ) and (Y,p 2 /q 2 ) maximal monotone maps, respectively. Then the orbit of x 1 under w = fg is given by x 1, fg(x 1 )=x 4, (fg) 2 (x 1 )=x 7,... as depicted in figure 1. Since x 7 = x 1 + 3, we have rot(fg)=3/2. The reader should at this point convince him or herself that Example 6.10 is not a particularly special easy case changing the input data or the word w might result in a longer computation, but will never be more technically challenging. We suggest the following: Exercise (computing rotation numbers) 1. With the input data of Example 6.10, compute rot(fg 2 ). 2. Change the input data of Example 6.10 so that the points of X and Y are ordered...x 1 <x 2 <y 1 <y 2 <x 3 <x 4 <... and verify that the (X, 1/2) and (Y,1/2) maximal monotone maps satisfy rot(fg) = 1. 26

27 3. Change the input data of Example 6.10 again so that p 2 /q 2 =1/3, and Y is Z-periodic of cardinality 3. (There are multiple possibilities for the ordering of points of X and Y in R, choose one). Compute rot(fg). Now choose a di erent ordering of the points in X and Y, and compute rot(fg) again. Monotone maps can always be approximated by homeomorphisms, as stated in the next proposition. Proposition Let f i be (X i,p i /q i ) maximal monotone maps. Then there exist g i 2 Homeo Z (R) withx i a periodic orbit for g i, and so that rot(w(g 1,...g n )) = rot(w(f 1,...f n )) holds for all w. Proof. Let X = S i X i as before. Pick much smaller than the minimum distance between any two points in X, and choose homeomorphisms g i so that g i (x,x+ ) (f i (x),f i (x)+ ) holds for all x 2 X. By Proposition 6.9, there exists x 2 X and m, k 2 Z such that w(f 1,...,f n ) pm (x) =x + pk for all p, and hence w(g 1,...,g n ) pm (x) 2 (x + pk Thus, rot(w(g 1,...g n )) = rot(w(f 1,...f n )) = m/k.,x+ pk + ). Combining Propositions 6.8 and 6.12 gives an e ective tool for computing the maximum possible value of rot(w(f 1,...f n )) when f i are homeomorphisms or monotone maps with periodic orbits X i. We summarize this as an explicit algorithm. Algorithm (The Calegari Walker algorithm) Input: 1. A word w in n letters 2. Rational numbers s i = p i /q i 2 Q, for i =1,...n. 3. A choice of p i /q i -periodic sets X i of cardinality q i. (The output of the algorithm depends only on the linear ordering of S i X i, or equivalently, its cyclic order after projection to S 1 ). Process: Let f i be a (X i,p i /q i ) maximal monotone map. Compute rot(w(f 1,...,f n )) by finding a periodic orbit as in Proposition 6.8. Output: The result is the maximum value of rot(w(g 1,...g n )), where g i are any monotone maps (equivalently, any elements of Homeo Z (R)) with periodic sets X i and rot(g i )=p i /q i. Variation: To compute R w (s 1,...s n ), take p i /q i to be in lowest terms, run the algorithm over all possible configurations of linear orderings on the union of the X i and take the maximum output. Note that, in the special case where rot(f i ) = 0, one can e ectively run the algorithm where X i is any finite subset of fix(f i ) this is allowed by viewing a cardinality q subset of fix(f i ) as a 0/q periodic orbit for f i.in[42] one can find several worked examples computing the maximum value of rot(f 1 f 2...f n ) given constraints on the fixed sets of f i. 6.2 Applications As an application, Algorithm 6.13 can be used to give a complete closed-form answer to Problem 5.16 for the word fg, and significant information in other cases. We start with a general lemma bounding the denominator of R w. Lemma Suppose s i are rational, that s 1 = p/q, and that w is a word in f 1,...f n. If R w (s 1,...s n )=n/m in lowest terms, then m apple q. 27

28 Proof. Apply Algorithm Since w ends in f 1, and f 1 (X) X 1, a closed orbit for w be a subset of X 1,henceisaZ-periodic set of cardinality m, for some m apple q. Remark As rot(w) is invariant under cyclic permutations of w (this follows from conjugation-invariance), Lemma 6.14 shows that the denominator of R w (s 1,...s n ) is bounded by min{q s i = p/q}. Theorem 6.16 (Theorem 3.9 in [15]). For any s, t 2 R, R fg (s, t) = p 1 + p 2 +1 sup p 1/qapples, p 2/qapplet q where p i, q are integers, q>0. The proof uses a lemma, which we leave as an exercise. Lemma For any integers p, k, and q, wehaver fg ( p q, k q ) are Z-periodic sets of cardinality q, ordered as p+k+1 q. In fact, if X and Y...x 1 <y 1 <x 2 <y 2 <x 3... then the composition of the (X, p/q) and (Y, k/q) maximal monotone maps has rotation number (p + k + 1)/q. Given this lemma, we outline the strategy of the proof of Theorem Proof outline. By Proposition 6.2, it su ces to prove the formula for rational s and t. Given s = p 1 /q 1 and t = p 2 /q 2,letX and Y be Z-periodic sets of cardinality q 1 and q 2 respectively, with f max and g max the maximal (X, p 1 /q 1 ) and (Y,p 2 /q 2 ) monotone maps. Suppose that the configuration of X and Y is such that rot(f max g max ) is maximized. By Lemma 6.14, rot(f max g max ) is rational of the form n/m, (and hence there is a n/m periodic orbit) for some n apple min{q 1,q 2 }. Using this, Calegari and Walker give a construction for re-ordering the points of X and Y, putting them into a standard form, without a ecting this periodic orbit. This is the technical work of the proof. From this standard form, one can read o the estimates p 1 /q 1 l/m and p 2 /q 2 (n l 1)/m for some l. Hence Hence R fg (p 1 /q 1,p 2 /q 2 )=n/m = l+(n l 1)+1 m, which is in the desired form. Commutators and Milnor Wood. In the study of surface group actions, we are particularly concerned with commutators. The Calegari Walker algorithm does not apply directly to a commutator as it is a word involving inverses; however, Theorem 6.16 can still be used to give an answer. Lemma 6.18 (Example 4.9 in [15]). Let f,g 2 Homeo Z (R). i) If rot(f) /2 Q or rot(g) /2 Q, then rot[f,g] = 0. ii) If rot(f) or rot(g) is of the form p/q, wherep/q 2 Q is in lowest terms (so q>0), then rot[f,g] apple 1/q. Proof. We present an argument slightly di erent from that in [15]. For the rational case, suppose rot(f) = p/q; by composing with translations (which does not a ect the value of [f,g]) we may assume 0 apple p/q < 1. Then rot(gf 1 g 1 )= p/q; 28

29 We apply Theorem 6.16: If j/m apple p/q and k/m apple p/q, with strict inequality in some case, then j+k j+k+1 m < 0, and since j and k are integers, we have m apple 0. If equality holds, then the maximum value is taken when j/m = p/q is in lowest terms, where p p +1 q =1/q, which implies that rot[f,g] apple 1/q. For the irrational case, suppose first that f is conjugate to translation T. After conjugacy we may assume f(x) =x +, and so there exists some x such that gf 1 g 1 (x) =x (this is easy if gf 1 g 1 (x) <x for all x, thenrot(gf 1 g 1 ) <, the same argument works if gf 1 g 1 (x) >x ). Thus, x is a fixed point for [f,g], and rot[f,g] = 0. In general, one can take a sequence f i of C 2 di eomorphisms that C 0 approximate f, i.e. such that f i (x)! f(x) and f 1 i (x)! f 1 (x) for all x 2 R. By continuity of rotation number, rot(f i )! rot(f), and we may take a subsequence such that rot(f i ) are either rational with denominator q i >i, in which case rot[f i,g] < 1/i, or rot(f i ) /2 Q. In this irrational case, Denjoy s theorem (which we hinted at in Section 3, but see e.g. [50, Ch. 3] for a precise statement) implies that each f i is conjugate to an irrational rotation, so rot([f i,g]) = 0. Continuity of rot applied to this sequence gives rot[f,g] = 0. As a further illustration of the power of the algorithmic technique, we give a quick proof of the Milnor Wood inequality. Theorem (Milnor Wood inequality, equivalent formulation) Let 2 Hom( g, Homeo(S 1 )). Then Y rot [ (ai ), (b i )] apple 2g 2. Proof. By Lemma 6.18, rot ([ (a i ), (b i )]) apple 1. Theorem 6.16 now gives the upper bound rot ([ (a 1 ), (b 1 )][ (a 2 ), (b 2 )]) apple 3, and inductively, rot! gy 1 [ (a i ), (b i )] apple 2g 3. i=1 Since Q i [ (a i), (b i )] is an integer translation, it follows from Lemma 5.14 that rot! gy [ (a i ), (b i )] = rot i=1! gy 1 [ (a i ), (b i )] + rot ([ (a g ), (b g )]) apple 2g 2. i=1 Remark In fact, Theorem 6.16 does more than give the estimate! gy 1 rot [ (a i ), (b i )] apple 2g 3, i=1 it implies that equality is achieved only if rot ([ (a i ), (b i )]) = 1. We ll use this fact in the next section. Understanding R w (s, t). We conclude this section with a short remark on the interesting problem of understanding the function R w itself. Figure 2 shows the plot of R w (s, t) the word w = fgffg,withthes axis on the left side, and t axis on the right. Because of its stairstep nautre (which all graphs of R w (s, t) share), Calegari and Walker call the graph a ziggurat. 29

30 Figure 2: A 3-D plot of R fgffg (s, t), 0 apple s, t < 1, from [15] Theorem 3.11 in [15] gives a more precise description of the stairstep nature of the ziggurat, as well as a faster algorithm to produce the graph. There are many open questions for instance, where exactly do the jumps occur and what values are taken at these points? (See the definition of slippery points and the slippery conjecture in [15]). Recent progress on describing self-similarity phenomena in ziggurats and the length of certain steps along the edges was made by A. Gordenko in [33] and furthered by S. Chowdhury [16]. 7 Rigidity of geometric representations In this section, we explain how to use rotation number coordinates and the Calegari Walker algorithm to prove the geometricity implies full rigidity Theorem Matsumoto s rigidity theorem As a warm-up and first case, we give a short proof of Corollary 4.16 on maximal representations without using Matsumoto s Theorem Proposition 7.1 (Corollary 4.16, equivalent formulation). Suppose 0 is a representation with rot ( Q [ 0 (a i ), 0 (b i )]) = 2g 2. Then 0 is fully rigid. The proof, though much easier than that of Theorem 4.17, is very much in the same spirit. We begin with an easy lemma used in both cases. Lemma 7.2 (Work in coordinates.). Let be any group, and suppose for each 2 the function 7! rot( ( )) is constant on a connected component C of Hom( g, Homeo(S 1 )). Then C consists of a single semi-conjugacy class. Proof. Let be as defined in Theorem 5.11, and fix and 0 in.sincerot( ( )) and rot( ( 0 )) are constant on C, it follows from the definition of that ( ( ), ( 0 )) is constant also. By 30